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(3x+2)*ln^3(x)
The derivative of the function/ (3x+2)*ln^3(x)

Derivative of (3x+2)*ln^3(x)

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
             3   
(3*x + 2)*log (x)
(3x+2)log⁡(x)3\left(3 x + 2\right) \log{\left(x \right)}^{3} 
d /             3   \
--\(3*x + 2)*log (x)/
dx
ddx(3x+2)log⁡(x)3\frac{d}{d x} \left(3 x + 2\right) \log{\left(x \right)}^{3}
Transform
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}

    f(x)=3x+2f{\left(x \right)} = 3 x + 2; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate 3x+23 x + 2 term by term:

      1. The derivative of a constant times a function is the constant times the derivative of the function.

        1. Apply the power rule: xx goes to 11

        So, the result is: 33

      2. The derivative of the constant 22 is zero.

      The result is: 33

    g(x)=log⁡(x)3g{\left(x \right)} = \log{\left(x \right)}^{3}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=log⁡(x)u = \log{\left(x \right)}.

    2. Apply the power rule: u3u^{3} goes to 3u23 u^{2}

    3. Then, apply the chain rule. Multiply by ddxlog⁡(x)\frac{d}{d x} \log{\left(x \right)}:

      1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

      The result of the chain rule is:

      3log⁡(x)2x\frac{3 \log{\left(x \right)}^{2}}{x}

    The result is: 3log⁡(x)3+3⋅(3x+2)log⁡(x)2x3 \log{\left(x \right)}^{3} + \frac{3 \cdot \left(3 x + 2\right) \log{\left(x \right)}^{2}}{x}

  2. Now simplify:

    3(xlog⁡(x)+3x+2)log⁡(x)2x\frac{3 \left(x \log{\left(x \right)} + 3 x + 2\right) \log{\left(x \right)}^{2}}{x}

The answer is:

3(xlog⁡(x)+3x+2)log⁡(x)2x\frac{3 \left(x \log{\left(x \right)} + 3 x + 2\right) \log{\left(x \right)}^{2}}{x}

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
                 2             
     3      3*log (x)*(3*x + 2)
3*log (x) + -------------------
                     x
3log⁡(x)3+3⋅(3x+2)log⁡(x)2x3 \log{\left(x \right)}^{3} + \frac{3 \cdot \left(3 x + 2\right) \log{\left(x \right)}^{2}}{x}
Simplify
The second derivative [src] 
  /           (-2 + log(x))*(2 + 3*x)\       
3*|6*log(x) - -----------------------|*log(x)
  \                      x           /       
---------------------------------------------
                      x
3⋅(6log⁡(x)−(3x+2)(log⁡(x)−2)x)log⁡(x)x\frac{3 \cdot \left(6 \log{\left(x \right)} - \frac{\left(3 x + 2\right) \left(\log{\left(x \right)} - 2\right)}{x}\right) \log{\left(x \right)}}{x}
Simplify
The third derivative [src] 
  /                                      /       2              \\
  |                          2*(2 + 3*x)*\1 + log (x) - 3*log(x)/|
3*|-9*(-2 + log(x))*log(x) + ------------------------------------|
  \                                           x                  /
------------------------------------------------------------------
                                 2                                
                                x
3(−9(log⁡(x)−2)log⁡(x)+2⋅(3x+2)(log⁡(x)2−3log⁡(x)+1)x)x2\frac{3 \left(- 9 \left(\log{\left(x \right)} - 2\right) \log{\left(x \right)} + \frac{2 \cdot \left(3 x + 2\right) \left(\log{\left(x \right)}^{2} - 3 \log{\left(x \right)} + 1\right)}{x}\right)}{x^{2}}
Simplify
The graph
Derivative of (3x+2)*ln^3(x)
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